Encoding Atomic Categories: Rendering It Strictly Directed

This paper solves the directed counterpart of a problem addressed in Language in Action (Van Benthem 1991: 108–9). There it is observed that LP derivability in an atomic goal category can be mimicked by LP derivability using one atomic category only. The abbreviation LP refers to the non-directed Lambek calculus with Permutation, a system which has also become known as the Lambek-Van Benthem calculus, and the result is due to Ponse (1988). In the present paper we will show that—a generalization of—this result can be extended to the directed system L, i.e., the associative calculus that was introduced in Lambek (1958): L derivability in any category can be mimicked by L derivability using one atomic category only.